- : unit = ()
- : unit = ()
h : heuristic = <heuristic>
- : unit = ()
APPLY CRITERIA (Marked dependency pairs)

TRS termination of:
[1] le(0,Y) -> true
[2] le(s(X),0) -> false
[3] le(s(X),s(Y)) -> le(X,Y)
[4] app(nil,Y) -> Y
[5] app(cons(N,L),Y) -> cons(N,app(L,Y))
[6] low(N,nil) -> nil
[7] low(N,cons(M,L)) -> iflow(le(M,N),N,cons(M,L))
[8] iflow(true,N,cons(M,L)) -> cons(M,low(N,L))
[9] iflow(false,N,cons(M,L)) -> low(N,L)
[10] high(N,nil) -> nil
[11] high(N,cons(M,L)) -> ifhigh(le(M,N),N,cons(M,L))
[12] ifhigh(true,N,cons(M,L)) -> high(N,L)
[13] ifhigh(false,N,cons(M,L)) -> cons(M,high(N,L))
[14] quicksort(nil) -> nil
[15] quicksort(cons(N,L)) ->
 app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))


Sub problem: 
guided: 

DP termination of:
<Marked_le(s(X),s(Y)) , Marked_le(X,Y)>
<Marked_app(cons(N,L),Y) , Marked_app(L,Y)>
<Marked_low(N,cons(M,L)) , Marked_iflow(le(M,N),N,cons(M,L))>
<Marked_low(N,cons(M,L)) , Marked_le(M,N)>
<Marked_iflow(true,N,cons(M,L)) , Marked_low(N,L)>
<Marked_iflow(false,N,cons(M,L)) , Marked_low(N,L)>
<Marked_high(N,cons(M,L)) , Marked_ifhigh(le(M,N),N,cons(M,L))>
<Marked_high(N,cons(M,L)) , Marked_le(M,N)>
<Marked_ifhigh(true,N,cons(M,L)) , Marked_high(N,L)>
<Marked_ifhigh(false,N,cons(M,L)) , Marked_high(N,L)>
<Marked_quicksort(cons(N,L)) , Marked_app(quicksort(low(N,L)),
                                cons(N,quicksort(high(N,L))))>
<Marked_quicksort(cons(N,L)) , Marked_quicksort(low(N,L))>
<Marked_quicksort(cons(N,L)) , Marked_low(N,L)>
<Marked_quicksort(cons(N,L)) , Marked_quicksort(high(N,L))>
<Marked_quicksort(cons(N,L)) , Marked_high(N,L)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 5 components:

{  <Marked_quicksort(cons(N,L)) , Marked_quicksort(high(N,L))> 
      --> <Marked_quicksort(cons(N,L)) , Marked_quicksort(high(N,L))>
  <Marked_quicksort(cons(N,L)) , Marked_quicksort(high(N,L))> 
     --> <Marked_quicksort(cons(N,L)) , Marked_quicksort(low(N,L))>
  <Marked_quicksort(cons(N,L)) , Marked_quicksort(low(N,L))> 
     --> <Marked_quicksort(cons(N,L)) , Marked_quicksort(high(N,L))>
  <Marked_quicksort(cons(N,L)) , Marked_quicksort(low(N,L))> 
     --> <Marked_quicksort(cons(N,L)) , Marked_quicksort(low(N,L))>
}

{  <Marked_app(cons(N,L),Y) , Marked_app(L,Y)> 
      --> <Marked_app(cons(N,L),Y) , Marked_app(L,Y)>
}

{  <Marked_iflow(false,N,cons(M,L)) , Marked_low(N,L)> 
      --> <Marked_low(N,cons(M,L)) , Marked_iflow(le(M,N),N,cons(M,L))>
  <Marked_iflow(true,N,cons(M,L)) , Marked_low(N,L)> 
     --> <Marked_low(N,cons(M,L)) , Marked_iflow(le(M,N),N,cons(M,L))>
  <Marked_low(N,cons(M,L)) , Marked_iflow(le(M,N),N,cons(M,L))> 
     --> <Marked_iflow(false,N,cons(M,L)) , Marked_low(N,L)>
  <Marked_low(N,cons(M,L)) , Marked_iflow(le(M,N),N,cons(M,L))> 
     --> <Marked_iflow(true,N,cons(M,L)) , Marked_low(N,L)>
}

{  <Marked_ifhigh(false,N,cons(M,L)) , Marked_high(N,L)> 
      --> <Marked_high(N,cons(M,L)) , Marked_ifhigh(le(M,N),N,cons(M,L))>
  <Marked_ifhigh(true,N,cons(M,L)) , Marked_high(N,L)> 
     --> <Marked_high(N,cons(M,L)) , Marked_ifhigh(le(M,N),N,cons(M,L))>
  <Marked_high(N,cons(M,L)) , Marked_ifhigh(le(M,N),N,cons(M,L))> 
     --> <Marked_ifhigh(false,N,cons(M,L)) , Marked_high(N,L)>
  <Marked_high(N,cons(M,L)) , Marked_ifhigh(le(M,N),N,cons(M,L))> 
     --> <Marked_ifhigh(true,N,cons(M,L)) , Marked_high(N,L)>
}

{  <Marked_le(s(X),s(Y)) , Marked_le(X,Y)> 
      --> <Marked_le(s(X),s(Y)) , Marked_le(X,Y)>
}
APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  le(0,Y) >= true ;
  le(s(X),0) >= false ;
  le(s(X),s(Y)) >= le(X,Y) ;
  app(nil,Y) >= Y ;
  app(cons(N,L),Y) >= cons(N,app(L,Y)) ;
  low(N,nil) >= nil ;
  low(N,cons(M,L)) >= iflow(le(M,N),N,cons(M,L)) ;
  iflow(true,N,cons(M,L)) >= cons(M,low(N,L)) ;
  iflow(false,N,cons(M,L)) >= low(N,L) ;
  high(N,nil) >= nil ;
  high(N,cons(M,L)) >= ifhigh(le(M,N),N,cons(M,L)) ;
  ifhigh(true,N,cons(M,L)) >= high(N,L) ;
  ifhigh(false,N,cons(M,L)) >= cons(M,high(N,L)) ;
  quicksort(nil) >= nil ;
  quicksort(cons(N,L)) >= app(quicksort(low(N,L)),cons(N,quicksort(high(N,L)))) ;
  Marked_quicksort(cons(N,L)) >= Marked_quicksort(low(N,L)) ;
  Marked_quicksort(cons(N,L)) >= Marked_quicksort(high(N,L)) ;
 }
 + Disjunctions:{ 
{
  Marked_quicksort(cons(N,L)) > Marked_quicksort(low(N,L)) ;
 }
{
  Marked_quicksort(cons(N,L)) > Marked_quicksort(high(N,L)) ;
 }
 } 
=== TIMER virtual : 10.000000 ===
Entering poly_solver
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 10.000000 ===
=== STOPING TIMER real ===
Sat solver returned
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
No solution found for these parameters.
Entering rpo_solver
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== STOPING TIMER virtual ===
constraint: le(0,Y) >= true
constraint: le(s(X),0) >= false
constraint: le(s(X),s(Y)) >= le(X,Y)
constraint: app(nil,Y) >= Y
constraint: app(cons(N,L),Y) >= cons(N,app(L,Y))
constraint: low(N,nil) >= nil
constraint: low(N,cons(M,L)) >= iflow(le(M,N),N,cons(M,L))
constraint: iflow(true,N,cons(M,L)) >= cons(M,low(N,L))
constraint: iflow(false,N,cons(M,L)) >= low(N,L)
constraint: high(N,nil) >= nil
constraint: high(N,cons(M,L)) >= ifhigh(le(M,N),N,cons(M,L))
constraint: ifhigh(true,N,cons(M,L)) >= high(N,L)
constraint: ifhigh(false,N,cons(M,L)) >= cons(M,high(N,L))
constraint: quicksort(nil) >= nil
constraint: quicksort(cons(N,L)) >= app(quicksort(low(N,L)),
                                     cons(N,quicksort(high(N,L))))
constraint: Marked_quicksort(cons(N,L)) >= Marked_quicksort(low(N,L))
constraint: Marked_quicksort(cons(N,L)) >= Marked_quicksort(high(N,L))
APPLY CRITERIA (Subterm criterion)

ST: Marked_app -> 1

APPLY CRITERIA (Subterm criterion)

ST: Marked_iflow -> 3
Marked_low -> 2

APPLY CRITERIA (Subterm criterion)

ST: Marked_ifhigh -> 3
Marked_high -> 2

APPLY CRITERIA (Subterm criterion)

ST: Marked_le -> 1

APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 0 components:
SOLVED
{
 
TRS termination of:
[1] le(0,Y) -> true
[2] le(s(X),0) -> false
[3] le(s(X),s(Y)) -> le(X,Y)
[4] app(nil,Y) -> Y
[5] app(cons(N,L),Y) -> cons(N,app(L,Y))
[6] low(N,nil) -> nil
[7] low(N,cons(M,L)) -> iflow(le(M,N),N,cons(M,L))
[8] iflow(true,N,cons(M,L)) -> cons(M,low(N,L))
[9] iflow(false,N,cons(M,L)) -> low(N,L)
[10] high(N,nil) -> nil
[11] high(N,cons(M,L)) -> ifhigh(le(M,N),N,cons(M,L))
[12] ifhigh(true,N,cons(M,L)) -> high(N,L)
[13] ifhigh(false,N,cons(M,L)) -> cons(M,high(N,L))
[14] quicksort(nil) -> nil
[15] quicksort(cons(N,L)) ->
 app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))
 ,
 CRITERION: MDP
 [ {
      
     DP termination of:
     <Marked_le(s(X),s(Y)) , Marked_le(X,Y)>
<Marked_app(cons(N,L),Y) , Marked_app(L,Y)>
<Marked_low(N,cons(M,L)) , Marked_iflow(le(M,N),N,cons(M,L))>
<Marked_low(N,cons(M,L)) , Marked_le(M,N)>
<Marked_iflow(true,N,cons(M,L)) , Marked_low(N,L)>
<Marked_iflow(false,N,cons(M,L)) , Marked_low(N,L)>
<Marked_high(N,cons(M,L)) , Marked_ifhigh(le(M,N),N,cons(M,L))>
<Marked_high(N,cons(M,L)) , Marked_le(M,N)>
<Marked_ifhigh(true,N,cons(M,L)) , Marked_high(N,L)>
<Marked_ifhigh(false,N,cons(M,L)) , Marked_high(N,L)>
<Marked_quicksort(cons(N,L)) , Marked_app(quicksort(low(N,L)),
                                cons(N,quicksort(high(N,L))))>
<Marked_quicksort(cons(N,L)) , Marked_quicksort(low(N,L))>
<Marked_quicksort(cons(N,L)) , Marked_low(N,L)>
<Marked_quicksort(cons(N,L)) , Marked_quicksort(high(N,L))>
<Marked_quicksort(cons(N,L)) , Marked_high(N,L)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_quicksort(cons(N,L)) , Marked_quicksort(low(N,L))>
<Marked_quicksort(cons(N,L)) , Marked_quicksort(high(N,L))>
 ,
 CRITERION: ORD
 [ 
     Solution found:
     RPO with AFS = AFS: low -> 1high -> 1iflow -> 2ifhigh -> 2
     and precedence:
     prec (All symbols are Lex.): { true  <  le ; le  >  true ; le  >  false ;
                                   false  <  le ; app  >  cons ;
                                   app  <  quicksort ; cons  <  app ;
                                   cons  <  quicksort ; quicksort  >  app ;
                                   quicksort  >  cons ; }

 ]} 
{
 
DP termination of:
<Marked_app(cons(N,L),Y) , Marked_app(L,Y)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_low(N,cons(M,L)) , Marked_iflow(le(M,N),N,cons(M,L))>
<Marked_iflow(true,N,cons(M,L)) , Marked_low(N,L)>
<Marked_iflow(false,N,cons(M,L)) , Marked_low(N,L)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
     <Marked_low(N,cons(M,L)) , Marked_iflow(le(M,N),N,cons(M,L))>
 ,
 CRITERION: SG
 [  ]} 
 ]} 
{
 
DP termination of:
<Marked_high(N,cons(M,L)) , Marked_ifhigh(le(M,N),N,cons(M,L))>
<Marked_ifhigh(true,N,cons(M,L)) , Marked_high(N,L)>
<Marked_ifhigh(false,N,cons(M,L)) , Marked_high(N,L)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
     <Marked_high(N,cons(M,L)) , Marked_ifhigh(le(M,N),N,cons(M,L))>
 ,
 CRITERION: SG
 [  ]} 
 ]} 
{
 
DP termination of:
<Marked_le(s(X),s(Y)) , Marked_le(X,Y)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
 ]} 
 ]} 

Cime worked for 0.256672 seconds (real time)
Cime Exit Status: 0